In Section  we perform the preliminary formal asymptotic analysis for simple eigenvalues using the method of matched asymptotic
expansions, cf. [1]}, [2]}, [3]}, [4]} for two scale asymptotic expansions. In Section 
we derive error estimates in the case of simple eigenvalues which will help us to detect open gaps
after a much more thorough analysis of multiple eigenvalues. The
perturbation of crossing dispersion curves require serious modifications of the standard asymptotic procedures because we can
no longer deal with a fixed Floquet parameter but we must investigate
the asymptotic behavior of the eigenvalues in a neighborhood of each
truss node, i.e., with the Floquet parameter in a certain short
interval. Recalling an idea from paper [5]}, in Section  we
introduce a fast Floquet parameter to describe this behavior and detect, in
different situations, open spectral gaps of width \(O(\varepsilon )\) ,
cf. Figure REF  a)–b), which appear due to splitting of the nodes
marked with \(\circ \)  and \(\mbox{\tiny $\square $}\)  in Figure REF .
This involves the characterization
of the projections of the shaded rectangles on the ordinate axis in
Figure REF , which represent the narrow gaps.
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